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PREFACE
This book presents some of the main ideas of game theory. It is designed to serve as a textbook for a one-semester
graduate course consisting of about 28 meetings each of 90 minutes.
The topics that we cover are those that we personally wo

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The notion of a strategic game encompasses situations much more complex than those described in the last five
examples. The following are representatives of three families of games that have been studied extensively:
auctions, games of timing, and

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There is a continuum of citizens, each of whom has a favorite position; the distribution of favorite positions is
given by a density function f on [0,1] with f(x) > 0 for all
. A candidate attracts the votes of those citizens
whose favorite positi

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Lemma 20.1
(Kakutani's fixed point theorem) Let X be a compact convex subset of
for which
for all
and let
be a set-valued function
the set f(x) is nonempty and convex
the graph of f is closed (i.e. for all sequences cfw_xn and cfw_yn such that
ha

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theorem to prove that there is an action
such that
is a Nash equilibrium of the game. (Such an
equilibrium is called a symmetric equilibrium.) Give an example of a finite symmetric game that has only
asymmetric equilibria.
2.5 Strictly Competitive

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In words, a maxminimizer for player i is an action that maximizes the payoff that player i can guarantee. A
maxminimizer for player 1 solves the problem maxx miny u1(x, y) and a maxminimizer for player 2 solves the
problem maxy minx = u2 (x, y).
I

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for all
and hence
for all
maxx miny u1(x, y) and x* is a maxminimizer for player 1.
, so that
. Thus u1(x*,y*) =
An analogous argument for player 2 establishes that y* is a maxminimzer for player 2 and u2(x*,y*) = maxy minx u2
(x, y), so that u1(x

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player l's payoff is at most her equilibrium payoff. In a game that is not strictly competitive a player's equilibrium
strategy does not in general have these properties (consider, for example, BoS (Figure 16.1).
Exercise 24.1
Let G be a strictly

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realized assigns to each state
the probability
if
and the probability zero otherwise
(i.e. the probability of conditional on
. As an example, if i() = for all
then player i has full
information about the state of nature. Alternatively, if
and for

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Figure 17.1
The Prisoner's Dilemma (Example 16.2)
Figure 17.2
Hawk-Dove (Example 16.3).
each animal is that in which it acts like a hawk while the other acts like a dove; the worst outcome is that in which
both animals act like hawks. Each animal

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Figure 16.1
Bach or Stravinsky? (BoS) (Example 15.3).
Figure 16.2
A coordination game (Example 16.1).
Example 16.1
(A coordination game) As in BoS, two people wish to go out together, but in this case they agree on the more
desirable concert. A g

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he chooses , given that every other player j chooses his equilibrium action
deviate, given the actions of the other players.
The following restatement of the definition is sometimes useful. For any
player i's best actions given a-i:
. Briefly, no

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The main interactions between the chapters. The areas of the boxes in
which the names of the chapters appear are proportional to the lengths
of the chapters. A solid arrow connecting two boxes indicates that one
chapter depends on the other; a do

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Exercises
Many of the exercises are challenging; we often use exercises to state subsidiary results. Instructors will probably
want to assign additional straightforward problems and adjust (by giving hints) the level of our exercises to make it

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to generic individuals. "They" has many merits as a singular pronoun, although its use can lead to ambiguities (and
complaints from editors). My preference is to use "she" for all individuals. Obviously this usage is not genderneutral, but its us

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and Tel Aviv University are gratefully appreciated. Special thanks are due to my friend Asher Wolinsky for endless
illuminating conversations. Part of my work on the book was supported by the United States-Israel Binational
Science Foundation (gra

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1
Introduction
1.1 Game Theory
Game theory is a bag of analytical tools designed to help us understand the phenomena that we observe when
decision-makers interact. The basic assumptions that underlie the theory are that decision-makers pursue welld

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ical; in principle a book could be written that had essentially the same content as this one and was devoid of
mathematics. A mathematical formulation makes it easy to define concepts precisely, to verify the consistency of
ideas, and to explore th

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devoted to noncooperative games; it does not express our evaluation of the relative importance of the two branches.
In particular, we do not share the view of some authors that noncooperative models are more "basic" than
cooperative ones; in our op

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the agents' activities. In a competitive analysis of this situation we look for a level of pollution consistent with the
actions that the agents take when each of them regards this level as given. By contrast, in a game theoretic analysis
of the si

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To model decision-making under uncertainty, almost all game theory uses the theories of von Neumann and
Morgenstern (1944) and of Savage (1972). That is, if the consequence function is stochastic and known to the
decision-maker (i.e. for each
the c